A note on equivalent theorem for Beta operators (Q2460797)

The Beta operators are given by \[ (B_nf)(x)=\sum\limits_{k=0}\limits^{\infty}p_{n,k}(x)f\left(\frac {k}{n+1}\right), \] where \[ p_{n,k}(x)=\frac{1}{n}\,\frac{\Gamma(n+k+1)}{\Gamma(n)\Gamma(k+1)}\frac{x^k}{(1+x)^{n+k+1}}, x\in [0,\infty). \] In this paper the author gives an equivalent theorem concerning on the weight space and norm \(\|f\|_N=\sup\limits_{[0,\infty)}|\frac{1}{1+x^N}f(x)|\). Both the direct and converse theorems are derived.